Do characters need to do a check to use items like, for example, the Rope of Knots, or can anyone use them?
Most of the time anyone can use wondrous items
The Magic Items section on Wondrous Items under the heading Activation says
Wondrous items are usually use-activated or activated by a command word, but details vary from item to item.
Command Word: If the activation is on command or if no activation method is suggested either in the magic item description or by the nature of the item, assume that a command word is needed to activate it. Command word activation means that a character speaks the word and the item activates. No other special knowledge is needed.
A command word can be a real word, but when this is the case, the holder of the item runs the risk of activating the item accidentally by speaking the word in normal conversation. More often, the command word is some nonsensical word, or a word or phrase from an ancient language. Activating a command word magic item is a standard action and does not provoke attacks of opportunity.
Sometimes the command word to activate an item is written right on the item. Occasionally, it might be hidden within a pattern or design engraved, carved, or built into the item, or the item might bear a clue to the command word.
The Knowledge (arcana) and Knowledge (history) skills can be useful in identifying command words or deciphering clues regarding them. In such cases, a successful DC 30 check is needed to come up with the word itself. If that check is failed, succeeding on a second check (DC 25) might provide a clue. The spells detect magic, identify, and analyze dweomer all reveal command words if the properties of the item are successfully identified.
Also, hidden in the section on Intelligent Wondrous Items, the game says that even a non-intelligent wondrous item has a 30% chance that
Something (a design, inscription, or the like) provides a clue to [an item's] function (or if the item is command word-activated, that it has the command word or words written on it).
Thus if the item's possessor knows the command word, the possessor can take a standard action to employ the wondrous item.
Further, a creature can employ the skill Use Magic Device (DC 25) to activate blindly a command-word-activated magic item for which he does't know the command word.
It's possible, though, that the wondrous item might be easier to use than that because some are
Use-Activated: This type of item merely has to be used in order to activate it. A character has to drink a potion, swing a sword, interpose a shield to deflect a blow, look through a lens, sprinkle dust, wear a ring, or don a hat. Use activation is generally straightforward and self-explanatory.
Many use-activated items are objects that a character wears. Continually functioning items are practically always items that one wears. A few must simply be in the character's possession (meaning on her person). However, some items made for wearing must still be activated. Although this activation sometimes requires a command word (see above), usually it means mentally willing the activation. The description of an item states whether a command word is needed in such a case.
However, this is Pathfinder, so exceptions will exist as to which creatures can use certain items, but those exceptions will either be listed in the item's description or specify a different activation method. The rope of knots is not an exception, functioning as it does "upon command," a standard action.